Specifications

University of Pretoria etd – Combrinck, M (2006)
strategy allows for both 1D and 3D model considerations. The late time mathematical
approximations are implemented for most procedures.
2.3 Analytical TDEM responses for four common models
Analytical responses for TDEM data are calculated by solving Maxwell’s equations. The
differential forms of the equations are valid at points in space only, but comparatively easy
to calculate, while the integral forms are valid at boundaries of units, more difficult to
calculate and used primarily to generate boundary conditions.
Table 2-1 Maxwell’s equations (Compiled from Ward and Hohmann (1988) and Kaufman and
Keller (1983)).
Maxwell’s equations in integral form: Maxwell’s equations in differential form:
qd
S
∫
=⋅ SD
∫
=⋅
S
d 0SB
surface through Flux=⋅=Φ
Φ
−=⋅
∫
∫
SB
lE
d
dt
d
d
S
B
B
S
D
IlH d
t
d
S
⋅
∂
∂
+=⋅
∫∫
ρ
=
⋅
∇
D
0
=
⋅
∇
B
t∂
∂
−=×∇
B
E
t∂
∂
+=×∇
D
jH
Maxwell’s equations are uncoupled differential equations that need to be coupled using the
constitutive equations that contain all the electrical and magnetic properties of the medium
through which EM propagation occurs (i.e. the earth in geophysics). The constitutive
equations are given by Ward and Hohmann (1988) as:
.Eσj
HµB
EεD
⋅=
⋅=
⋅=
[F/m] typermittividielectric
[H/m] typermeabilimagnetic[S/m]ty conductivi :
[Coulomb] charge [C/mdensity chargeelectric :
]
2
[A/mdensitycurrent[A]current
[A/m]intensity fieldmagnetic ][C/mntdisplaceme dielectric
[Tesla]inductionmagnetic [V/m]intensity fieldelectric
3
2
:,
:,,
:]
::
::
::
ε
µσ
jI
HD
BE
ε
µσ
ρ
q
5